Solved: Are the statements in 4654 true or false Give anexplanation for your answer.If

Show that y = xex + 2 is a solution of the initial value problem dy dx = (1 x)e x, y(0) = 2.

Show that y = sin(2t), for 0 t < /4, is a solution to the initial value problem dy dt = 21 y2, y(0) = 0.

In Exercises 38, find the general solution to the differential equation. dy dx = 2x

In Exercises 38, find the general solution to the differential equation. dy dt = t 2

In Exercises 38, find the general solution to the differential equation. dy dx = x3 + 5x4

In Exercises 38, find the general solution to the differential equation. dy dt = et

In Exercises 38, find the general solution to the differential equation. dy dx = cos x

In Exercises 38, find the general solution to the differential equation. dy dx = 1 x, where x > 0

In Exercises 912, find the solution of the initial value problem. dy dx = 3x2 , y(0) = 5

In Exercises 912, find the solution of the initial value problem. dy dx = x5 + x6 , y(1) = 2

In Exercises 912, find the solution of the initial value problem. dy dx = ex, y(0) = 7

In Exercises 912, find the solution of the initial value problem. dy dx = sin x, y(0) = 3

A rock is thrown downward with velocity 10 ft/sec from a bridge 100 ft above the water. How fast is the rock going when it hits the water?

A water balloon launched from the roof of a building at time t = 0 has vertical velocity v(t) = 32t + 40 feet/sec at time t seconds, with v > 0 corresponding to upward motion. (a) If the roof of the building is 30 feet above the ground, find an expression for the height of the water balloon above the ground at time t. (b) What is the average velocity of the balloon between t = 1.5 and t = 3 seconds? (c) A 6-foot person is standing on the ground. How fast is the water balloon falling when it strikes the person on the top of the head?

A car starts from rest at time t = 0 and accelerates at 0.6t + 4 meters/sec2 for 0 t 12. How long does it take for the car to go 100 meters?

Ice is forming on a pond at a rate given by dy dt = k t, where y is the thickness of the ice in inches at time t measured in hours since the ice started forming, and k is a positive constant. Find y as a function of t.

A revenue R(p) is obtained by a farmer from selling grain at price p dollars/unit. The marginal revenue is given by R (p) = 25 2p. (a) Find R(p). Assume the revenue is zero when the price is zero. (b) For what prices does the revenue increase as the price increases? For what prices does the revenue decrease as price increases?

A firms marginal cost function is MC = 3q2 + 6q + 9. (a) Write a differential equation for the total cost, C(q). (b) Find the total cost function if the fixed costs are 400.

A tomato is thrown upward from a bridge 25 m above the ground at 40 m/sec. (a) Give formulas for the acceleration, velocity, and height of the tomato at time t. (b) How high does the tomato go, and when does it reach its highest point? (c) How long is it in the air?

A car going 80 ft/sec (about 55 mph) brakes to a stop in five seconds. Assume the deceleration is constant. (a) Graph the velocity against time, t, for 0 t 5 seconds. (b) Represent, as an area on the graph, the total distance traveled from the time the brakes are applied until the car comes to a stop. (c) Find this area and hence the distance traveled. (d) Now find the total distance traveled using antidifferentiation.

An object is shot vertically upward from the ground with an initial velocity of 160 ft/sec. (a) At what rate is the velocity decreasing? Give units. (b) Explain why the graph of velocity of the object against time (with upward positive) is a line. (c) Using the starting velocity and your answer to part (b), find the time at which the object reaches the highest point. (d) Use your answer to part (c) to decide when the object hits the ground. (e) Graph the velocity against time. Mark on the graph when the object reaches its highest point and when it lands. (f) Find the maximum height reached by the object by considering an area on the graph. (g) Now express velocity as a function of time, and find the greatest height by antidifferentiation.

A stone thrown upward from the top of a 320-foot cliff at 128 ft/sec eventually falls to the beach below. (a) How long does the stone take to reach its highest point? (b) What is its maximum height? (c) How long before the stone hits the beach? (d) What is the velocity of the stone on impact?

A 727 jet needs to attain a speed of 200 mph to take off. If it can accelerate from 0 to 200 mph in 30 seconds, how long must the runway be? (Assume constant acceleration.)

A cat, walking along the window ledge of a New York apartment, knocks off a flower pot, which falls to the street 200 feet below. How fast is the flower pot traveling when it hits the street? (Give your answer in ft/sec and in mph, given that 1 ft/sec = 15/22 mph.)

An Acura NSX going at 70 mph stops in 157 feet. Find the acceleration, assuming it is constant.

(a) Find the general solution of the differential equation dy/dx = 2x + 1. (b) Sketch a graph of at least three solutions. (c) Find the solution satisfying y(1) = 5. Graph this solution with the others from part (b).

(a) Find and graph the general solution of the differential equation dy dx = sin x + 2. (b) Find the solution of the initial value problem dy dx = sin x + 2, y(3) = 5.

On the moon, the acceleration due to gravity is about 1.6 m/sec2 (compared to g 9.8 m/sec2 on earth). If you drop a rock on the moon (with initial velocity 0), find formulas for: (a) Its velocity, v(t), at time t. (b) The distance, s(t), it falls in time t.

(a) Imagine throwing a rock straight up in the air. What is its initial velocity if the rock reaches a maximum height of 100 feet above its starting point? (b) Now imagine being transplanted to the moon and throwing a moon rock vertically upward with the same velocity as in part (a). How high will it go? (On the moon, g = 5 ft/sec2.)

An object is dropped from a 400-foot tower. When does it hit the ground and how fast is it going at the time of the impact?

The object in Problem 30 falls off the same 400-foot tower. What would the acceleration due to gravity have to be to make it reach the ground in half the time?

A ball that is dropped from a window hits the ground in five seconds. How high is the window? (Give your answer in feet.)

On the moon the acceleration due to gravity is 5 ft/sec2. An astronaut jumps into the air with an initial upward velocity of 10 ft/sec. How high does he go? How long is the astronaut off the ground?

. A particle of mass, m, acted on by a force, F, moves in a straight line. Its acceleration, a, is given by Newtons Law: F = ma. The work, W, done by a constant force when the particle moves through a displacement, d, is W = F d. The velocity, v, of the particle as a function of time, t, is given in Figure 6.25. What is the sign of the work done during each of the time intervals: [0, t1], [t1, t2], [t2, t3], [t3, t4],[t2, t4]?

Assume the acceleration of a moving body is g and its initial velocity and position are v0 and s0 respectively. Find velocity, v, and position, s, as a function of t.

Galileo was the first person to show that the distance traveled by a body falling from rest is proportional to the square of the time it has traveled, and independent of the mass of the body. Derive this result from the fact that the acceleration due to gravity is a constestant.

While attempting to understand the motion of bodies under gravity, Galileo stated that: The time in which any space is traversed by a body starting at rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest velocity and the velocity just before acceleration began. (a) Write Galileos statement in symbols, defining all the symbols you use. (b) Check Galileos statement for a body dropped off a 100-foot building accelerating from rest under gravity until it hits the ground. (c) Show why Galileos statement is true in general.

Newtons law of gravity says that the gravitational force between two bodies is attractive and given by F = GMm r2 , where G is the gravitational constant, m and M are the masses of the two bodies, and r is the distance between them. This is the famous inverse square law. For a falling body, we take M to be the mass of the earth and r to be the distance from the body to the center of the earth. So, actually, r changes as the body falls, but for anything we can easily observe (say, a ball dropped from the Tower of Pisa), it wont change significantly over the course of the motion. Hence, as an approximation, it is reasonable to assume that the force is constant. According to Newtons second law, acceleration is caused by a force and Force = Mass Acceleration. (a) Find the differential equation for the position, s of a moving body as a function of time. (b) Explain how the differential equation shows the acceleration of the body is independent of its mass.

In his Dialogues Concerning Two New Sciences, Galileo wrote: The distances traversed during equal intervals of time by a body falling from rest stand to one another in the same ratio as the odd numbers beginning with unity. Assume, as is now believed, that s = (gt2)/2, where s is the total distance traveled in time t, and g is the acceleration due to gravity. (a) How far does a falling body travel in the first second (between t = 0 and t = 1)? During the second second (between t = 1 and t = 2)? The third second? The fourth second? (b) What do your answers tell you about the truth of Galileos statement?

In Problems 4043, explain what is wrong with the statement. A rock dropped from a 400-foot cliff takes twice as long to hit the ground as it would if it were dropped from a 200-foot cliff.

In Problems 4043, explain what is wrong with the statement. The function y = cos(t 2) is a solution to the initial value problem dy dt = sin(t 2 ), y(0) = 1.

In Problems 4043, explain what is wrong with the statement. Two solutions to a differential equation dy/dx = f(x) have graphs which cross at the initial value.

In Problems 4043, explain what is wrong with the statement. A differential equation cannot have a constant solution.

In Problems 4445, give an example of: Two different solutions to the differential equation dy dt = t + 3.

In Problems 4445, give an example of: A differential equation that has solution y = cos (5x).

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If F(x) is an antiderivative of f(x), then y = F(x) is a solution to the differential equation dy/dx = f(x).

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If y = F(x) is a solution to the differential equation dy/dx = f(x), then F(x) is an antiderivative of f(x).

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If an object has constant nonzero acceleration, then the position of the object as a function of time is a quadratic polynomial.

Are the statements in Problems 4654 true or false? Give an explanation for your answer. In an initial value problem for the differential equation dy/dx = f(x), the value of y at x = 0 is always specified

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If f(x) is positive for all x, then there is a solution of the differential equation dy/dx = f(x) where y(x) is positive for all x.

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If f(x) > 0 for all x then every solution of the differential equation dy/dx = f(x) is an increasing function.

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If two solutions of a differential equation dy/dx = f(x) have different values at x = 3 then they have different values at every x.

Are the statements in Problems 4654 true or false? Give an explanation for your answer. If the function y = f(x) is a solution of the differential equation dy/dx = sin x/x, then the function y = f(x)+5 is also a solution.

Are the statements in Problems 4654 true or false? Give an explanation for your answer. There is only one solution y(t) to the initial value problem dy/dt = 3t 2, y(1) = .